Nuprl Lemma : non-trivial-open-box
∀[X:CubicalSet]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀bx:open_box(X;I;J;x;i). ∀y:nameset(J). ∀c:ℕ2.
(¬(filter(λf.((dimension(f) =z y) ∧b (direction(f) =z c));bx)
= []
∈ ({f:{f:I-face(X;I)| (f ∈ bx)} | ↑((dimension(f) =z y) ∧b (direction(f) =z c))} List)))
Proof
Definitions occuring in Statement :
open_box: open_box(X;I;J;x;i),
face-direction: direction(f),
face-dimension: dimension(f),
I-face: I-face(X;I),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
l_member: (x ∈ l),
filter: filter(P;l),
nil: [],
list: T List,
band: p ∧b q,
int_seg: {i..j-},
assert: ↑b,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
set: {x:A| B[x]} ,
lambda: λx.A[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
open_box: open_box(X;I;J;x;i),
prop: ℙ,
subtype_rel: A ⊆r B,
nameset: nameset(L),
coordinate_name: Cname,
int_upper: {i...},
int_seg: {i..j-},
implies: P ⇒ Q,
not: ¬A,
false: False,
and: P ∧ Q,
uimplies: b supposing a,
so_apply: x[s],
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
face-name: face-name(f),
pi2: snd(t),
pi1: fst(t),
guard: {T},
lelt: i ≤ j < k,
sq_stable: SqStable(P),
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
face-direction: direction(f),
face-dimension: dimension(f),
so_lambda: λ2x.t[x],
sq_type: SQType(T)
Lemmas referenced :
filter_type,
I-face_wf,
l_member_wf,
band_wf,
eq_int_wf,
face-dimension_wf,
face-direction_wf,
list-subtype,
list_wf,
assert_wf,
non_null_filter,
int_seg_wf,
not_wf,
null_wf3,
subtype_rel_list,
top_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
equal_wf,
assert_of_null,
equal-wf-T-base,
nameset_wf,
open_box_wf,
coordinate_name_wf,
cubical-set_wf,
iff_transitivity,
select_wf,
int_seg_properties,
length_wf,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
iff_weakening_uiff,
assert_of_band,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
nameset_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
lambdaEquality,
applyEquality,
because_Cache,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
productElimination,
natural_numberEquality,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
unionElimination,
equalityElimination,
dependent_functionElimination,
addLevel,
impliesFunctionality,
independent_pairFormation,
baseClosed,
dependent_pairFormation,
imageMemberEquality,
imageElimination,
int_eqEquality,
intEquality,
computeAll,
productEquality,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}bx:open\_box(X;I;J;x;i). \mforall{}y:nameset(J). \mforall{}c:\mBbbN{}2.
(\mneg{}(filter(\mlambda{}f.((dimension(f) =\msubz{} y) \mwedge{}\msubb{} (direction(f) =\msubz{} c));bx) = []))
Date html generated:
2017_10_05-AM-10_20_43
Last ObjectModification:
2017_07_28-AM-11_21_03
Theory : cubical!sets
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